Group Structure
In this section we will forget about the topological structure of the circle group and look only at its structure as an abstract group.
The circle group T is a divisible group. Its torsion subgroup is given by the set of all nth roots of unity for all n, and is isomorphic to Q/Z. The structure theorem for divisible groups tells us that T is isomorphic to the direct sum of Q/Z with a number of copies of Q. The number of copies of Q must be c (the cardinality of the continuum) in order for the cardinality of the direct sum to be correct. But the direct sum of c copies of Q is isomorphic to R, as R is a vector space of dimension c over Q. Thus
The isomorphism
can be proved in the same way, as C× is also a divisible abelian group whose torsion subgroup is the same as the torsion subgroup of T.
Read more about this topic: Circle Group
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